This invention relates to determination of strains developed by cyclic deformation of structures such as pneumatic tires.
Analysis of the tire temperature rise is reported in the published scientific and patent literature. Such analysis can be found, for example, in the following articles:
P. Kainradl, G. Kaufmann and F. Schmidt, in Kautschuk Und Gummi-Kunststoffe, vol. 19, 27, (1966), discussing the relationship between temperature rise in pneumatic tires and viscoelastic properties of rubber;
J. M. Collins, W. L. Jackson and P. S. Oubridge, "Relevance of Elastic and Loss Moduli of Tyre Components to Tyre Energy Loss", in Transactions of the Rubber Industry, vol. 40 T239 (1964).
U.S. Pat. No. 3,553,307 to F. J. Kovac and G. W. Rye, "Treatment of Polyester Tire Cord", col. 5, lines 22-54.
When such cyclic dynamic experiments are conducted using a sinusoidal alteration of stress or strain, for so-called "linear" viscoelastic solids, the instantaneous stress (.sigma.) varies periodically with time (t) according to EQU .sigma.(t) = .epsilon..sub.o (E' sin .omega.t + E" cos .omega.t) (1)
Sin(.omega.E' is the dynamic modulus, E" is the loss modulus; .epsilon..sub.o is the strain amplitude; and .omega. is the frequency of cyclic straining in radians/sec. The variation of instantaneous stress with time during cyclic stressing can also be expressed for linear viscoelastic solids by means of the amplitude of the stress, .sigma..sub.o, and the phase angle .delta. between the stress and strain: EQU .sigma.(t) = .sigma..sub.o sin(.omega.t + .delta.) = .sigma..sub.o cos .delta. sin(.omega. t) + .sigma..sub.o sin .delta. cos (.omega.t) (2)
Consequently, ##EQU1## The important assumption in the linear viscoelastic theory is that the moduli E', E" and the phase angle .delta. are constant during the cycle.